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Relational Calculus Logic, like whiskey, loses its beneficial effect when taken in too large quantities. --Lord Dunsany Relational Calculus • Query has the form: {T | p(T)} – p(T) is a formula containing T • Answer = tuples T for which p(T) = true. Formulae • Atomic formulae: T Relation T.a op T.b T.a op constant … op is one of ,, ,,, • A formula can be: – – – – an atomic formula p, pq, pq,pq R( p(R)) R( p(R)) Free and Bound Variables • Quantifiers: and • Use of X or X binds X. – A variable that is not bound is free. • Recall our definition of a query: – {T | p(T)} • Important restriction: — T must be the only free variable in p(T). — all other variables must be bound using a quantifier. Simple Queries • Find all sailors with rating above 7 {S |S Sailors S.rating > 7} • Find names and ages of sailors with rating above 7. {S | S1 Sailors(S1.rating > 7 S.sname = S1.sname S.age = S1.age)} – Note: S is a variable of 2 fields (i.e. S is a projection of Sailors) Joins Find sailors rated > 7 who’ve reserved boat #103 { S | SSailors S.rating > 7 R(RReserves R.sid = S.sid R.bid = 103) } Joins (continued) Find sailors rated > 7 who’ve reserved a red boat { S | SSailors S.rating > 7 R(RReserves R.sid = S.sid B(BBoats B.bid = R.bid B.color = ‘red’)) } • This may look cumbersome, but it’s not so different from SQL! Universal Quantification Find sailors who’ve reserved all boats { S | SSailors BBoats (RReserves (S.sid = R.sid B.bid = R.bid)) } A trickier example… Find sailors who’ve reserved all Red boats in existence… { S | SSailors B Boats ( B.color = ‘red’ R(RReserves S.sid = R.sid B.bid = R.bid)) } Alternatively… { S | SSailors B Boats ( B.color ‘red’ R(RReserves S.sid = R.sid B.bid = R.bid)) } a b is the same as a b b T a F T T F F T T A Remark: Unsafe Queries • syntactically correct calculus queries that have an infinite number of answers! Unsafe queries. – e.g., S | S Sailors – Solution???? Don’t do that! Your turn … • Schema: Movie(title, year, studioName) ActsIn(movieTitle, starName) Star(name, gender, birthdate, salary) • Queries to write in Relational Calculus: 1. Find all movies by Paramount studio 2. … movies whose stars are all women 3. … movies starring Kevin Bacon 4. Find stars who have been in a film w/Kevin Bacon 5. Stars within six degrees of Kevin Bacon* 6. Stars connected to K. Bacon via any number of films** * Try two degrees for starters ** Good luck with this one! Answers … 1. Find all movies by Paramount studio {M | MMovie M.studioName = ‘Paramount’} Answers … 2. Movies whose stars are all women {M | MMovie AActsIn((A.movieTitle = M.title) SStar(S.name = A.starName S.gender = ‘F’))} Answers … 3. Movies starring Kevin Bacon {M | MMovie AActsIn(A.movieTitle = M.title A.starName = ‘Bacon’))} Answers … 4. Stars who have been in a film w/Kevin Bacon {S | SStar AActsIn(A.starName = S.name A2ActsIn(A2.movieTitle = A.movieTitle A2.starName = ‘Bacon’))} S: name … A: A2: movie movie star star ‘Bacon’ Answers … two 5. Stars within six degrees of Kevin Bacon {S | SStar AActsIn(A.starName = S.name A2ActsIn(A2.movieTitle = A.movieTitle A3ActsIn(A3.starName = A2.starName A4ActsIn(A4.movieTitle = A3.movieTitle A4.starName = ‘Bacon’))} Two degrees: S: name … A: movie star A2: movie star A3: movie star A4: movie star ‘Bacon’ Answers … 6. Stars connected to K. Bacon via any number of films • Sorry … that was a trick question – Not expressible in relational calculus!! • What about in relational algebra? – We will be able to answer this question shortly … Expressive Power • Expressive Power (Theorem due to Codd): – Every query that can be expressed in relational algebra can be expressed as a safe query in relational calculus; the converse is also true. • Relational Completeness: Query language (e.g., SQL) can express every query that is expressible in relational algebra/calculus. (actually, SQL is more powerful, as we will see…) Question: • Can we express query #6 in relational algebra? • A: If we could, then by Codd’s theorem we could also express it in relational calculus. However, we know the latter is not possible, so the answer is no. Summary • Formal query languages — simple and powerful. – Relational algebra is operational • used as internal representation for query evaluation plans. – Relational calculus is “declarative” • query = “what you want”, not “how to compute it” – Same expressive power --> relational completeness. • Several ways of expressing a given query – a query optimizer should choose the most efficient version.